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In this paper, we consider numerical approximations for the viscousCahn-Hilliard equation with hyperbolic relaxation. This type of equationsprocesses energy-dissipative structure. The main challenge in solving such adiffusive system numerically is how to develop high order temporaldiscretization for the hyperbolic and nonlinear terms, allowing largetime-marching step, while preserving the energy stability, i.e. the energydissipative structure at the time-discrete level. We resolve this issue bydeveloping two second-order time-marching schemes using the recently developed"Invariant Energy Quadratization" approach where all nonlinear terms arediscretized semi-explicitly. In each time step, one only needs to solve asymmetric positive definite (SPD) linear system. All the proposed schemes arerigorously proven to be unconditionally energy stable, and the second-orderconvergence in time has been verified by time step refinement testsnumerically. Various 2D and 3D numerical simulations are presented todemonstrate the stability, accuracy and efficiency of the proposed schemes.

Linear, Second order and Unconditionally Energy stable schemes for The  Viscous Cahn-Hilliard Equation with hyperbolic relaxation using the invariant  energy quadratization method